\(MA(q)\)
From last time - a \(MA(q)\) process with \(w_t \overset{iid}{\sim} N(0,\sigma^2_w)\), \[y_t = \delta + w_t + \theta_1 \, w_{t-1} + \theta_2 \, w_{t-2} + \cdots + \theta_q \, w_{t-q} \]
has the following properties, \[E(y_t) = \delta\]
\[
Var(y_t) = \gamma(0) = (1 + \theta_1^2 + \theta_2^2 + \cdots + \theta_q^2) \, \sigma_w^2
\]
\[
Cov(y_t, y_{t+h}) = \gamma(h) = \begin{cases}
\sigma^2_w \sum_{j=0}^{q-|h|} \theta_j \theta_{j+|h|} & \text{if $|h| \leq q$} \\
0 & \text{if $|h| > q$}
\end{cases}
\]
and is stationary for any values of \((\theta_1, \ldots, \theta_q)\)
\(MA(\infty)\)
If we let \(q \to \infty\) then process will be stationary if and only if the moving average coefficients (\(\theta\) ’s) are square summable, i.e.
\[ \sum_{i=1}^\infty \theta_i^2 < \infty\]
which is necessary so that the \(Var(y_t) < \infty\) condition is met for weak stationarity.
Sometimes, a slightly stronger condition known as absolute summability, \(\sum_{i=1}^\infty |\theta_i| < \infty\) is necessary (e.g. for some CLT related asymptotic results).
Invertibility vs Stationarity
A \(MA(q)\) process is invertible if \(y_t = \delta + \theta_q(L) \, w_t\) can be rewritten as an exclusively \(AR\) process (of possibly infinite order), i.e. \(\phi(L) \, y_t = \alpha + w_t\).
\(~\)
Conversely, an \(AR(p)\) process is stationary if \(\phi_p(L) \, y_t = \delta + w_t\) can be rewritten as an exclusively \(MA\) process (of possibly infinite order), i.e. \(y_t = \delta + \theta(L) \, w_t\).
\(~\)
So using our results w.r.t. \(\phi(L)\) it follows that if all of the roots of \(\theta_q(L)\) are outside the complex unit circle then the moving average process is invertible.
